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Search: id:A132581
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| A132581 |
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Number of antichains in the first n elements of the infinite Boolean lattice. |
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+0
2
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| 1, 2, 3, 5, 6, 11, 14, 19, 20, 39, 53, 78, 84, 134, 148, 167, 168, 335, 483, 765, 849, 1466, 1681, 1988, 2008, 3700, 4414, 5489, 5573, 7265, 7413, 7580
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Every nonnegative integer represents a finite set of nonnegative integers, and conversely, by the mapping that takes n into the exponents of 2 in its binary representation. Thus 0 represents the empty set, and 9 represents {0,3}, etc.
The sequence counts the antichains in the partial ordering {0,1,...,n}, which is really the family of sets {emptyset,{0},{1},{0,1},{2},...}.
The sequence of differences, A132582, enumerates the antichains in the infinite Boolean lattice {0,1,2,...} whose largest element is n. For example, the five such antichains when n=6 are {6}, {1,6}, {3,6}, {5,6}, {3,5,6}.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.4 (in preparation).
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CROSSREFS
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Cf. A132582.
Sequence in context: A104012 A039037 A050049 this_sequence A039839 A039844 A038196
Adjacent sequences: A132578 A132579 A132580 this_sequence A132582 A132583 A132584
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KEYWORD
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nonn
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AUTHOR
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D. E. Knuth, Nov 18 2007
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